Subspace design[1,2] 

Also known as \(q\)-design, Geometric design.


A \(q\)-ary code that can be mapped into a subspace \(t\)-\((n,w,\lambda)_q\) design.

Subspace designs exist for all parameters in sufficiently large dimension that also satisfies divisibility constraints [3,4].


See [5] for a review on subspace designs.Popular summary of the existence of subspace designs in Quanta Magazine.



  • Combinatorial design — Combinatorial designs are designs on a space of fixed-weight binary strings (a.k.a. Johnson association scheme) [7,8]. Subspace designs reduce to combinatorial designs for \(q=2\).


Cameron, Peter J. "Generalisation of Fisher’s inequality to fields with more than one element." Combinatorics, London Math. Soc. Lecture Note Ser 13 (1973): 9-13.
V. Guruswami and C. Xing, “List decoding reed-solomon, algebraic-geometric, and gabidulin subcodes up to the singleton bound”, Proceedings of the forty-fifth annual ACM symposium on Theory of Computing (2013) DOI
A. Fazeli, S. Lovett, and A. Vardy, “Nontrivial t-designs over finite fields exist for all t”, Journal of Combinatorial Theory, Series A 127, 149 (2014) DOI
P. Keevash, A. Sah, and M. Sawhney, “The existence of subspace designs”, (2023) arXiv:2212.00870
M. Braun, M. Kiermaier, and A. Wassermann, “q-Analogs of Designs: Subspace Designs”, Network Coding and Subspace Designs 171 (2018) DOI
Ph. Delsarte, “Hahn Polynomials, Discrete Harmonics, andt-Designs”, SIAM Journal on Applied Mathematics 34, 157 (1978) DOI
Delsarte, Philippe. "An algebraic approach to the association schemes of coding theory." Philips Res. Rep. Suppl. 10 (1973): vi+-97.
V. I. Levenshtein, “Universal bounds for codes and designs,” in Handbook of Coding Theory 1, eds. V. S. Pless and W. C. Huffman. Amsterdam: Elsevier, 1998, pp.499-648.
Page edit log

Your contribution is welcome!

on (edit & pull request)— see instructions

edit on this site

Zoo Code ID: subspace_design

Cite as:
“Subspace design”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024.
@incollection{eczoo_subspace_design, title={Subspace design}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:

Cite as:

“Subspace design”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024.